Mean curvature self-shrinkers of high genus: Non-compact examples

Abstract

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in ${\mathbb{R}}^3$ over a $2\pi /(g+1)$-periodic graph on an equator of the unit sphere ${\mathbb{S}}^2\subseteq {\mathbb{R}}^3$, with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted Hölder spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

41 pages, 1 figure; minor typos fixed; to appear in J. Reine Angew. Math

References in corpus (1)

• Self-shrinkers with a rotational symmetry

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